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Local systems which do not come from abelian varieties

Paul Brommer-Wierig, Yeuk Hay Joshua Lam

TL;DR

The paper studies $ar{\boldsymbol{Q}}_\ell$-local systems of geometric origin on curves over a finite field and proves that, after removing finitely many points, there exist such local systems that do not arise from a family of abelian varieties. It develops a slope-based criterion for local systems to come from abelian varieties, modeled on Hodge-theoretic constraints, and applies filtered $\varphi$-module techniques to relate Frobenius slopes to possible origins. The mirror quintic (Dwork) family provides a concrete counterexample with generic $v$-slopes $0,1,2,3$ and monodromy $\,\mathfrak{sp}_{4}$, showing the criterion is violated. The general result follows by pulling back this counterexample along finite maps from any curve, yielding a local system of geometric origin on a punctured curve that does not come from abelian varieties. This demonstrates a fundamental distinction between the point case and higher-dimensional bases in the landscape of geometric local systems over finite fields and motivates further criteria for classifying abelian-origin local systems.

Abstract

For each smooth curve over a finite field, after puncturing it at finitely many points, we construct local systems on it of geometric origin which do not come from a family of abelian varieties. We do so by proving a criterion which must be satisfied by local systems which do come from abelian varieties, inspired by an analogous Hodge theoretic criterion in characteristic zero.

Local systems which do not come from abelian varieties

TL;DR

The paper studies -local systems of geometric origin on curves over a finite field and proves that, after removing finitely many points, there exist such local systems that do not arise from a family of abelian varieties. It develops a slope-based criterion for local systems to come from abelian varieties, modeled on Hodge-theoretic constraints, and applies filtered -module techniques to relate Frobenius slopes to possible origins. The mirror quintic (Dwork) family provides a concrete counterexample with generic -slopes and monodromy , showing the criterion is violated. The general result follows by pulling back this counterexample along finite maps from any curve, yielding a local system of geometric origin on a punctured curve that does not come from abelian varieties. This demonstrates a fundamental distinction between the point case and higher-dimensional bases in the landscape of geometric local systems over finite fields and motivates further criteria for classifying abelian-origin local systems.

Abstract

For each smooth curve over a finite field, after puncturing it at finitely many points, we construct local systems on it of geometric origin which do not come from a family of abelian varieties. We do so by proving a criterion which must be satisfied by local systems which do come from abelian varieties, inspired by an analogous Hodge theoretic criterion in characteristic zero.
Paper Structure (14 sections, 7 theorems, 16 equations)

This paper contains 14 sections, 7 theorems, 16 equations.

Table of Contents

  1. Introduction
  2. Statement of the result
  3. Discussions: Fq versus C
  4. Sketch of proof
  5. Notation
  6. Acknowledgments
  7. Reminder on l-adic local systems
  8. Tannakian categories
  9. Monodromy of l-adic local systems
  10. Proof of the result
  11. The criterion
  12. Filtered phi-modules
  13. The mirror quintic family violates the criterion
  14. General case

Key Result

Theorem 1.1.2

Let $C$ be a smooth projective curve over a finite field of characteristic $p$. Let $\ell$ be a prime number different from $p$. There exists a finite set of points $S\subseteq C$ and a $\overline{\mathbb Q}_\ell$-local system on $C\setminus S$, of geometric origin, which does not come from a family

Theorems & Definitions (22)

  • Example 1.1.1
  • Theorem 1.1.2
  • Definition 2.2.1
  • Lemma 3.1.1
  • proof
  • Proposition 3.1.2
  • proof
  • Remark 3.1.3
  • Definition 3.2.1
  • Remark 3.2.2
  • ...and 12 more